Algebraic Number Fields

Abstract

Let us consider the diophantine equation

$$x^2 - dy^2 = 1$$

((4.1))

, erroneously called Pell’s equation. (For its history, see Ref. 9.) Here d ≠ 0 is a square-free integer. We seek the integer solutions of (4.1). If d < 0, these solutions are (±1,0) for d < −1 and (±1,0), (0, ±1) for d = −1. However, if d > 1, it is a nontrivial fact that (4.1) has infinitely many solutions in integers. If we let G denote the set of these solutions, then G has a group structure (cf. Exercise 2.4). Moreover, up to multiplication by −1 [i.e., −(x, y) = (−x, −y)], G is an infinite cyclic group. A generator is a solution with the smallest |y ₁| (and hence the smallest |x ₁|) > 0.